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We consider the semilinear elliptic problem Ã¢ÂˆÂ’ÃŽÂ”u+u=ÃŽÂ»K(x)up+f(x) in ÃŽÂ©, u>0 in ÃŽÂ©, uÃ¢ÂˆÂˆH01(ÃŽÂ©), where ÃŽÂ»Ã¢Â‰Â¥0, NÃ¢Â‰Â¥3, 1<p<(N+2)/(NÃ¢ÂˆÂ’2), and ÃŽÂ© is an exterior strip domain in Ã¢Â„ÂN. Under some suitable conditions on K(x) and f(x), we show that there exists a positive constant ÃŽÂ»Ã¢ÂˆÂ— such that the above semilinear elliptic problem has at least two solutions if ÃŽÂ»Ã¢ÂˆÂˆ(0,ÃŽÂ»Ã¢ÂˆÂ—), a unique positive solution if ÃŽÂ»=ÃŽÂ»Ã¢ÂˆÂ—, and no solution if ÃŽÂ»>ÃŽÂ»Ã¢ÂˆÂ—. We also obtain some bifurcation results of the solutions at ÃŽÂ»=ÃŽÂ»Ã¢ÂˆÂ—

Existence and bifurcation for some elliptic problems on exterior strip domains